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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 5733.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5733.e1 | 5733g3 | \([1, -1, 1, -30659, -2058514]\) | \(37159393753/1053\) | \(90311725413\) | \([2]\) | \(9216\) | \(1.2039\) | |
5733.e2 | 5733g4 | \([1, -1, 1, -8609, 280550]\) | \(822656953/85683\) | \(7348698545643\) | \([2]\) | \(9216\) | \(1.2039\) | |
5733.e3 | 5733g2 | \([1, -1, 1, -1994, -29032]\) | \(10218313/1521\) | \(130450270041\) | \([2, 2]\) | \(4608\) | \(0.85732\) | |
5733.e4 | 5733g1 | \([1, -1, 1, 211, -2572]\) | \(12167/39\) | \(-3344878719\) | \([2]\) | \(2304\) | \(0.51075\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5733.e have rank \(1\).
Complex multiplication
The elliptic curves in class 5733.e do not have complex multiplication.Modular form 5733.2.a.e
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.